Abstract

In this article, we study and prove the new existence theorems of fixed points for contraction mappings in modular metric spaces.

AMS: 47H09; 47H10.

Keywords

modular metric spacesmodular spacescontraction mappingsfixed points

1 Introduction

Let (X, d) be a metric space. A mapping T : X → X is a contraction if

d(T(x),T(y))≤kd(x,y),

(1.1)

for all x, y∈X, where 0 ≤ k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 [1]. Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [2–10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11–13] and others. Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14–18] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4, 19–26].

In 2008, Chistyakov [27] introduced the notion of modular metric spaces generated by F-modular and develop the theory of this spaces, on the same idea he was defined the notion of a modular on an arbitrary set and develop the theory of metric spaces generated by modular such that called the modular metric spaces in 2010 [28].

In this article, we study and prove the existence of fixed point theorems for contraction mappings in modular metric spaces.

2 Preliminaries

We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14, 15, 27–29] for more details).

Definition 2.1. Let X be a vector space overℝ(orℂ). A functional ρ : X → [0, ∞] is called a modular if for arbitrary x and y, elements of X satisfies the following three conditions :

is called a modular space. Xρ is a vector subspace of X it can be equipped with an F-norm defined by setting

xρ=inf{λ>0:ρxλ≤λ},x∈Xρ.

(2.2)

In addition, if ρ is convex, then the modular space Xρ coincides with

Xρ*={x∈X:∃λ=λ(x)>0such thatρ(λx)<∞}

(2.3)

and the functional xρ*=inf{λ>0:ρxλ≤1}is an ordinary norm on Xρ* which is equivalence to xρ(see [16]).

Let X be a nonempty set, λ ∈ (0, ∞) and due to the disparity of the arguments, function w : (0, ∞) × X × X → [0, ∞] will be written as wλ(x, y) = w(λ, x, y) for all λ > 0 and x, y∈X.

Definition 2.2. [[28], Definition 2.1] Let X be a nonempty set. A function w : (0, ∞) × X × X → [0, ∞] is said to be a metric modular on X if satisfying, for all x, y, z∈X the following condition holds:

(i) wλ(x, y) = 0 for all λ > 0 if and only if x = y;

(ii) wλ(x, y) = wλ(y, x) for all λ > 0;

(iii) wλ + μ(x, y) ≤ wλ(x, z) + wμ(z, y) for all λ, μ > 0.

If instead of (i), we have only the condition

(i') wλ(x, x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on X.

The main property of a (pseudo) modular w on a set X is a following: given x, y∈X, the function 0 < λ↦wλ(x, y) ∈ [0, ∞] is a nonincreasing on (0, ∞).

In fact, if 0 < μ < λ, then (iii), (i') and (ii) imply

wλ(x,y)≤wλ-μ(x,x)+wμ(x,y)=wμ(x,y).

(2.4)

It follows that at each point λ > 0 the right limit wλ+0(x,y):=limε→+0wλ+ε(x,y) and the left limit wλ-0(x,y):=limε→+0wλ-ε(x,y) exists in [0, ∞] and the following two inequalities hold :

wλ+0(x,y)≤wλ(x,y)≤wλ-0(x,y).

(2.5)

Definition 2.3. [[28], Definition 3.3] A function w : (0, ∞) × X × X → [0, ∞] is said to be a convex (metric) modular on X if it is satisfies the conditions (i) and (ii) from Definition 2.2 as well as this condition holds;

(iv) wλ+μ(x,y)=λλ+μwλ(x,z)+μλ+μwμ(z,y)forallλ,μ>0andx,y,z∈X.

If instead of (i), we have only the condition (i') from Definition 2.2, then w is called a convex(metric) pseudomodular on X.

From [27, 28], we know that, if x0∈X, the set Xw={x∈X:limλ→∞wλ(x,x0)=0} is a metric space, called a modular space, whose metric is given by dw∘(x,y)=inf{λ>0:wλ(x,y)≤λ} for all x, y∈Xw. Moreover, if w is convex, the modular set Xwis equal to Xw*={x∈X:∃λ=λ(x)>0 such that wλ(x, x0) <∞} and metrizable by dw*(x,y)=inf{λ>0:wλ(x,y)≤1}for all x,y∈Xw*. We know that (see [[28], Theorem 3.11]) if X is a real linear space, ρ : X → [0, ∞] and

wλ(x,y)=ρx-yλfor all λ>0and x,y∈X,

(2.6)

then ρ is modular (convex modular) on X in the sense of (A.1)-(A.4) if and only if w is metric modular (convex metric modular, respectively) on X. On the other hand, if w satisfy the following two conditions (i) wλ(μx, 0) = wλ/μ (x, 0) for all λ, μ > 0 and x∈X, (ii) wλ(x + z, y + z) = wλ(x, y) for all λ > 0 and x, y, z∈X, if we set ρ(x) = w1(x, 0) with (2.6) holds, where x∈X, then

(i)

Xρ = Xw is a linear subspace of X and the functional xρ=dw∘(x,0), x∈Xρ, is an F-norm on Xρ;

(ii)

if w is convex, Xρ*≡Xw*(0)=Xρ is a linear subspace of X and the functional xρ=dw*(x,0),x∈Xρ*, is an norm on Xρ*.

Similar assertions hold if replace the word modular by pseudomodular. If w is metric modular in X, we called the set Xw is modular metric space.

By the idea of property in metric spaces and modular spaces, we defined the following:

Definition 2.4. Let Xwbe a modular metric space.

(1) The sequence (xn)n∈ℕin Xwis said to be convergent to x∈Xwif wλ(xn, x) → 0, as n → ∞ for all λ > 0.

(2) The sequence (xn) n∈ℕinXwis said to be Cauchy if wλ(xm, xn) → 0, as m, n → ∞ for all λ > 0.

(3) A subset C of Xwis said to be closed if the limit of a convergent sequence of C always belong to C.

(4) A subset C of Xwis said to be complete if any Cauchy sequence in C is a convergent sequence and its limit is in C.

(5) A subset C of Xwis said to be bounded if for all λ > 0 δw(C) = sup{wλ(x, y); x, y∈C} <∞.

3 Main results

In this section, we prove the existence of fixed points theorems for contraction mapping in modular metric spaces.

Definition 3.1. Let w be a metric modular on X and Xwbe a modular metric space induced by w and T : Xw→ Xwbe an arbitrary mapping. A mapping T is called a contraction if for each x, y∈Xwand for all λ > 0 there exists 0 ≤ k < 1 such that

wλ(Tx,Ty)≤kwλ(x,y).

(3.1)

Theorem 3.2. Let w be a metric modular on X and Xwbe a modular metric space induced by w. If Xwis a complete modular metric space and T : Xw→ Xwis a contraction mapping, then T has a unique fixed point in Xw. Moreover, for any x∈Xw, iterative sequence {Tnx} converges to the fixed point.

for all λ > 0 and for each n∈ℕ. Therefore, limn→∞wλ(xn+1,xn)=0 for all λ > 0. So for each λ > 0, we have for all ∊> 0 there exists n0∈ℕ such that wλ(xn, xn+1) <∊ for all n∈ℕ with n ≥ n0. Without loss of generality, suppose m, n∈ℕ and m > n. Observe that, for λm-n>0, there exists nλ/(m-n)∈ℕ such that

for all m, n ≥ nλ/(m-n). This implies {xn}n∈ℕis a Cauchy sequence. By the completeness of Xw, there exists a point x∈Xw such that xn→ × as n → ∞.

By the notion of metric modular w and the contraction of T, we get

wλ(Tx,x)≤wλ2(Tx,Txn)+wλ2(Txn,x)≤kwλ2(x,xn)+wλ2(xn+1,x)

(3.2)

for all λ > 0 and for each n∈ℕ. Taking n → ∞ in (3.2) implies that wλ(Tx, x) = 0 for all λ > 0 and thus Tx = x. Hence, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z is another fixed point of T. We see that

wλ(x,z)=wλ(Tx,Tz)≤kwλ(x,z)

for all λ > 0. Since 0 ≤ k < 1, we get wλ(x, z) = 0 for all λ > 0 this implies that x = z. Therefore, x is a unique fixed point of T and the proof is complete. □

Theorem 3.3. Let w be a metric modular on X and Xwbe a modular metric space induced by w. If Xwis a complete modular metric space and T : Xw→ Xwis a contraction mapping. Suppose x*∈Xwis a fixed point of T, {εn} is a sequence of positive numbers for whichlimn→∞εn=0, and {yn} ⊆Xwsatisfies

Now let ε > 0. Since limn→∞εn=0, there exists N∈ℕ such that for m ≥ N, εm≤ ε. Thus,

∑i=0mkm-iεi=∑i=0Nkm-iεi+∑i=N+1mkm-iεi≤km-N∑i=0NkN-iεi+ε∑i=N+1mkm-i.

(3.5)

Taking limit as m → ∞ in (3.5), we have

limm→∞∑i=0mkm-iεi=0.

(3.6)

Since x0 is a fixed point of T and using result of Theorem 3.2, we get the sequence {Tnx} converge to x*. This implies that

limm→∞wλ2(Tm+1x,x*)=0

(3.7)

for all λ > 0. From (3.4), (3.6) and (3.7), we have

limm→∞wλ(ym+1,x*)=0

(3.8)

for all λ > 0 which implies that limn→∞yn=x*. □

Theorem 3.4. Let w be a metric modular on X and Xwbe a modular metric space induced by w. If Xwis a complete modular metric space and T : Xw → Xwis a mapping, which TNis a contraction mapping for some positive integer N. Then, T has a unique fixed point in Xw.

Proof. By Theorem 3.2 , TN has a unique fixed point u∈Xw. From TN(Tu) = TN+1u = T(TNu) = Tu, so Tu is a fixed point of TN. By the uniqueness of fixed point of TN, we have Tu = u. Thus, u is a fixed point of T. Since fixed point of T is also fixed point of TN, we can conclude that T has a unique fixed point in Xw. □

for all λ > 0, where 0 ≤ k < 1. Then, T has a unique fixed point in Bw(x*, γ).

Proof. By Theorem 3.2 , we only prove that Bw(x*, γ) is complete and Tx∈Bw(x*, γ), for all x∈Bw(x*, γ). Suppose that {xn} is a Cauchy sequence in Bw(x*, γ), also {xn} is a Cauchy sequence in Xw. Since Xw is complete, there exists x∈Xw such that

limn→∞wλ2(xn,x)=0

(3.10)

for all λ > 0. Since for each n∈ℕ, xn∈Bw(x*, γ), using the property of metric modular, we get

wλ(x*,x)≤wλ2(x*,xn)+wλ2(xn,x)≤γ+wλ2(xn,x*)

(3.11)

for all λ > 0. It follows the inequalities (3.10) and (3.11), we have wλ(x*, x) ≤ γ which implies that x∈Bw(x*, γ). Therefore, {xn} is convergent sequence in Bw(x*, γ) and also Bw(x*, γ) is complete.

Next, we prove that Tx∈Bw(x*, γ) for all x∈Bw(x*, γ). Let x∈Bw(x*, γ). From the inequalities (3.9), using the contraction of T and the notion of metric modular, we have

for all x, y∈Xwand for all λ > 0, wherek∈[0,12), then Thas a unique fixed point in Xw. Moreover, for any x∈Xw, iterative sequence {Tnx} converges to the fixed point.

Proof. Let x0 be an arbitrary point in Xw and we write x1 = Tx0, x2 = Tx1 = T2x0, and in general, xn = Txn-1= Tnx0 for all n∈ℕ. If Txn0-1=Txn0for some n0∈ℕ, then Txn0=xn0. Thus, xn0 is a fixed point of T. Suppose that Txn-1≠ Txn for all n∈ℕ. For k∈[0,12), we have

for all λ > 0 and for all n∈ℕ. Put β:=k1-k, since k∈[0,12), we get β∈ [0, 1) and hence

wλ(xn+1,xn)≤βwλ(xn,xn-1)≤β2wλ(xn-1,xn-2)⋮≤βnwλ(x1,x0)

(3.15)

for all λ > 0 and for all n∈ℕ. Similar to the proof of Theorem 3.2, we can conclude that {xn} is a Cauchy sequence and by the completeness of Xw there exists a point x∈Xw such that xn → x as n → ∞. By the property of metric modular and the inequality (3.12), we have

Notes

Declarations

Acknowledgements

The authors thank the referee for comments and suggestions on this manuscript. The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0029/2553). The second author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for the Ph.D. Program. The third author was supported by the Commission on Higher Education and the Thailand Research Fund (Grant No.MRG5380044). Moreover, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC Project No. 54000267).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT)

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